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High Frequency Limit of the Helmholtz Equations

Jean-David Benamou 1 François Castella Thodoros Katsaounis Benoît Perthame 2
1 ONDES - Modeling, analysis and simulation of wave propagation phenomena
Inria Paris-Rocquencourt, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR2706
Abstract : We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not share the quadratic aspect) in the limit, then, the lack of L2 bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity.
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Submitted on : Wednesday, May 24, 2006 - 11:09:31 AM
Last modification on : Wednesday, May 11, 2022 - 12:06:04 PM
Long-term archiving on: : Sunday, April 4, 2010 - 11:25:56 PM


  • HAL Id : inria-00072875, version 1



Jean-David Benamou, François Castella, Thodoros Katsaounis, Benoît Perthame. High Frequency Limit of the Helmholtz Equations. [Research Report] RR-3785, INRIA. 1999. ⟨inria-00072875⟩



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